Contemporary Mathematics

# 8.5Percentiles

Figure 8.31 Two students graduating with the same class rank could be in different percentiles depending on the school population. (credit: "graduation caps" by John Walker/Flickr, CC BY 2.0)

## Learning Objectives

After completing this section, you should be able to:

1. Compute percentiles.
2. Solve application problems involving percentiles.

A college admissions officer is comparing two students. The first, Anna, finished 12th in her class of 235 people. The second, Brian, finished 10th in his class of 170 people. Which of these outcomes is better? Certainly 10 is less than 12, which favors Brian, but Anna’s class was much bigger. In fact, Anna beat out 223 of her classmates, which is $223235≈95%223235≈95%$ of her classmates, while Brian bested 160 out of 170 people, or 94%. Comparing the proportions of the data values that are below a given number can help us evaluate differences between individuals in separate populations. These proportions are called percentiles. If $p%p%$ of the values in a dataset are less than a number $nn$, then we say that $nn$ is at the $pp$th percentile.

## Finding Percentiles

There are some other terms that are related to "percentile" with meanings you may infer from their roots. Remember that the word percent means “per hundred.” This reflects that percentiles divide our data into 100 pieces. The word quartile has a root that means “four.” So, if a data value is at the first quantile of a dataset, that means that if you break the data into four parts (because of the quart-), this data value comes after the first of those four parts. In other words, it’s greater than 25% of the data, placing it at the 25th percentile. Quintile has a root meaning “five,” so a data value at the third quintile is greater than three-fifths of the data in the set. That would put it at the 60th percentile. The general term for these is quantiles (the root quant– means “number”).

In Mean, Median, and Mode, we defined the median as a number that is greater than no more than half of the data in a dataset and is less than no more than half of the data in the dataset. With our new term, we can more easily define it: The median is the value at the 50th percentile (or second quartile).

Let’s look at some examples.

## Example 8.28

### Finding Percentiles

Consider the dataset 5, 8, 12, 1, 2, 16, 2, 15, 20, 22.

1. At what percentile is the value 5?
2. What value is at the 60th percentile?

Consider the dataset 2, 5, 8, 16, 12, 1, 8, 6, 15, 4.
1.
What value is at the 80th percentile?
2.
At what percentile is the value 12?

In each of the examples above, the computations were made easier by the fact that the we were looking for percentiles that “came out evenly” with respect to the number of values in our dataset. Things don’t always work out so cleanly. Further, different sources will define the term percentile in different ways. In fact, Google Sheets has three built-in functions for finding percentiles, none of which uses our definition. Some of the definitions you’ll see differ in the inequality that is used. Ours uses “less than or equal to,” while others use “less than” (these correspond roughly to Google Sheets’ ‘PERCENTILE.INC’ and ‘PERCENTILE.EXC’). Some of them use different methods for interpolating values. (This is what we did when we first computed medians without technology; if there were an even number of data values in our dataset, found the mean of the two values in the middle. This is an example of interpolation. Most computerized methods use this technique.) Other definitions don’t interpolate at all, but instead choose the closest actual data value to the theoretical value. Fortunately, for large datasets, the differences among the different techniques become very small.

So, with all these different possible definitions in play, what will we use? For small datasets, if you’re asked to compute something involving percentiles without technology , use the technique we used in the previous example. In all other cases, we’ll keep things simple by using the built-in ‘PERCENTILE’ and ‘PERCENTRANK’ functions in Google Sheets (which do the same thing as the ‘PERCENTILE.INC’ and ‘PERCENTRANK.INC’ functions; they’re “inclusive, interpolating” definitions).

## Example 8.29

### Using Google Sheets to Compute Percentiles: Average SAT Scores

The data in “AvgSAT” contains the average SAT score for students attending every institution of higher learning in the US for which data is available.

1. What score is at the 3rd quartile?
2. What score is at the 40th percentile?
3. At what percentile is Albion College in Michigan (average SAT: 1132)?
4. At what percentile is Oregon State University (average SAT: 1205)?

Looking again at the “AvgSAT” dataset:
1.
What score is at the 15th percentile?
2.
What score is at the 90th percentile?
3.
At what percentile is the University of Missouri (Columbia campus), whose average SAT score is 1244?
4.
At what percentile is Rice University in Texas, whose average SAT score is 1513?

## Example 8.30

### Using Google Sheets to Compute Percentiles: In-State Tuition

The dataset "InState" contains the in-state tuitions of every college and university in the country that reported that data to the Department of Education. Use that dataset to answer these questions.

1. What tuition is at the second quintile?
2. What tuition is at the 95th percentile?
3. At what percentile is Walla Walla University in Washington (in-state tuition: $28,035)? 4. At what percentile is the College of Saint Mary in Nebraska (in-state tuition:$20,350)?

Looking again at the "InState" dataset, answer these questions.
1.
What tuition is at the 10th percentile?
2.
What tuition is at the fourth quintile?
3.
At what percentile is the main campus of New Mexico State University (in-state tuition: $6,686)? 4. At what percentile is Bowdoin College in Maine (in-state tuition:$53,922)?

Given the data 10, 12, 14, 18, 21, 23, 24, 25, 29, and 30, compute the following without technology:
32.
The value at the 30th percentile
33.
The value at the first quintile
34.
At what percentile 29 falls
35.
At what percentile 24 falls

For the remainder of these problems, use the dataset "MLB2019," which gives the number of wins for each Major League Baseball team in the 2019 season. Use Google Sheets to compute your answers.

36.
How many wins is at the 30th percentile?
37.
How many wins is at the 90th percentile?
38.
How many wins is at the first quartile?
39.
At what percentile are the Chicago Cubs (CHC, 84 wins)?
40.
At what percentile are the Tampa Bay Rays (TBR, 96 wins)?
41.
At what percentile are the Toronto Blue Jays (TOR, 67 wins)?

## Section 8.5 Exercises

For the following exercises, use the following twenty data values to answer the questions without technology: 1, 4, 6, 7, 12, 15, 21, 25, 29, 30, 31, 33, 39, 43, 44, 45, 51, 55, 60, 63
1 .
What data value is at the 10th percentile?
2 .
What data value is at the 55th percentile?
3 .
What data value is at the 90th percentile?
4 .
What data value is at the 30th percentile?
5 .
What data value is at the first quartile?
6 .
What data value is at the third quintile?
7 .
At what percentile is 29?
8 .
At what percentile is 55?
9 .
At what percentile is 4?
10 .
At what percentile is 51?
For the following exercises, use the data in "TNSchools", which has data on many institutions of higher education in the state of Tennessee. Here are what the columns represent:
Column Name Description
PTUG Proportion of undergraduates who attend part-time
InState Tuition and fees for in-state students
OutState Tuition and fees for out-of-state students
FacSal Mean monthly faculty salary
Pell Proportion of students receiving Pell Grants
MedDebt Median student loan debt at degree completion
StartAge Mean age at the time of entry
Female Proportion of students who identify as female
(source: https://data.ed.gov)
11 .
What admission rate is at the second quintile?
12 .
What admission rate is at the 80th percentile?
13 .
What admission rate is at the 90th percentile?
14 .
At what percentile is East Tennessee State University for admission rate?
15 .
At what percentile is Rhodes College for admission rate?
16 .
At what percentile is Freed-Hardeman University for admission rate?
17 .
What proportion of part-time undergraduate enrollment is at the third quartile?
18 .
What proportion of part-time undergraduate enrollment is at the 15th percentile?
19 .
What proportion of part-time undergraduate enrollment is at the 40th percentile?
20 .
At what percentile is Lee University for proportion of part-time undergraduate enrollment?
21 .
At what percentile is Fisk University for proportion of part-time undergraduate enrollment?
22 .
At what percentile is Middle Tennessee State University for proportion of part-time undergraduate enrollment?
23 .
What median student loan debt is at the 10th percentile?
24 .
What median student loan debt is at the fourth quintile?
25 .
What median student loan debt is at the 85th percentile?
26 .
At what percentile is Carson-Newman College for median student loan debt?
27 .
At what percentile is Austin Peay State University for median student loan debt?
28 .
At what percentile is Belmont University for median student loan debt?
29 .
What mean starting age is at the first quartile?
30 .
What mean starting age is at the 67th percentile?
31 .
What mean starting age is at the 35th percentile?
32 .
At what percentile is the University of the South for mean starting age?
33 .
At what percentile is Lincoln Memorial University for mean starting age?
34 .
At what percentile is the University of Tennessee-Chattanooga for mean starting age?
35 .
What proportion of students who identify as female is at the third quintile?
36 .
What proportion of students who identify as female is at the 12th percentile?
37 .
What proportion of students who identify as female is at the 85th percentile?
38 .
At what percentile is Martin Methodist College for proportion of students who identify as female?
39 .
At what percentile is Tennessee Technological University for proportion of students who identify as female?
40 .
At what percentile is Maryville College for proportion of students who identify as female?